The Experts below are selected from a list of 1002 Experts worldwide ranked by ideXlab platform

Visconti G. - One of the best experts on this subject based on the ideXlab platform.

  • CWENO: Uniformly accurate reconstructions for balance laws
    'American Mathematical Society (AMS)', 2018
    Co-Authors: Puppo G., Semplice M., Visconti G.
    Abstract:

    In this paper we introduce a general framework for defining and studying essentially nonoscillatory reconstruction procedures of arbitrarily high order of accuracy, interpolating data in the central stencil around a given computational cell (CWENO). This technique relies on the same selection mechanism of smooth stencils adopted in WENO, but here the pool of candidates for the selection includes Polynomials of different degrees. This seemingly minor difference allows us to compute the analytic expression of a Polynomial Interpolant, approximating the unknown function uniformly within a cell, instead of only at one point at a time. For this reason this technique is particularly suited for balance laws for finite volume schemes, when averages of source terms require high order quadrature rules based on several points; in the computation of local averages, during refinement in h-adaptive schemes; or in the transfer of the solution between grids in moving mesh techniques, and in general when a globally defined reconstruction is needed. Previously, these needs have been satisfied mostly by ENO reconstruction techniques, which, however, require a much wider stencil than the CWENO reconstruction studied here, for the same accuracy

  • CWENO: uniformly accurate reconstructions for balance laws
    'American Mathematical Society (AMS)', 2016
    Co-Authors: Puppo G., Semplice M., Visconti G.
    Abstract:

    In this paper we introduce a general framework for defining and studying essentially non-oscillatory reconstruction procedures of arbitrarily high order accuracy, interpolating data in a central stencil around a given computational cell ($\CWENO$). This technique relies on the same selection mechanism of smooth stencils adopted in $\WENO$, but here the pool of candidates for the selection includes Polynomials of different degrees. This seemingly minor difference allows to compute an analytic expression of a Polynomial Interpolant, approximating the unknown function uniformly within a cell, instead of only at one point at a time. For this reason this technique is particularly suited for balance laws for finite volume schemes, when averages of source terms require high order quadrature rules based on several points; in the computation of local averages, during refinement in h-adaptive schemes; or in the transfer of the solution between grids in moving mesh techniques, and in general when a globally defined reconstruction is needed. Previously, these needs have been satisfied mostly by ENO reconstruction techniques, which, however, require a much wider stencil then the $\CWENO$ reconstruction studied here, for the same accuracy

Vandewalle Stefan - One of the best experts on this subject based on the ideXlab platform.

  • Explicit barycentric weights for Polynomial interpolation in the roots or extrema of classical orthogonal Polynomials
    'American Mathematical Society (AMS)', 2014
    Co-Authors: Wang Haiyong, Huybrechs Daan, Vandewalle Stefan
    Abstract:

    © 2014 American Mathematical Society. Barycentric interpolation is arguably the method of choice for numerical Polynomial interpolation. The Polynomial Interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the interpolation points. Few explicit formulae for these barycentric weights are known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861-877], the authors have shown that the barycentric weights of the roots of Legendre Polynomials can be expressed explicitly in terms of the weights of the corresponding Gaussian quadrature rule. This idea was subsequently implemented in the Chebfun package [L. N. Trefethen and others, The Chebfun Development Team, 2011] and in the process generalized by the Chebfun authors to the roots of Jacobi, Laguerre and Hermite Polynomials. In this paper, we explore the generality of the link between barycentric weights and Gaussian quadrature and show that such relationships are related to the existence of lowering operators for orthogonal Polynomials. We supply an exhaustive list of cases, in which all known formulae are recovered and also some new formulae are derived, including the barycentric weights for Gauss-Radau and Gauss-Lobatto points. Based on a fast O(n) algorithm for the computation of Gaussian quadrature, due to Hale and Townsend, this leads to an O(n) computational scheme for barycentric weights.23 pages, 4 figures, revised version with minor changesstatus: publishe

  • Explicit barycentric weights for Polynomial interpolation in the roots or extrema of classical orthogonal Polynomials
    2012
    Co-Authors: Wang Haiyong, Huybrechs Daan, Vandewalle Stefan
    Abstract:

    Barycentric interpolation is arguably the method of choice for numerical Polynomial interpolation. The Polynomial Interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the interpolation points. Few explicit formulae for these barycentric weights are known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861--877], the authors have shown that the barycentric weights of the roots of Legendre Polynomials can be expressed explicitly in terms of the weights of the corresponding Gaussian quadrature rule. This idea was subsequently implemented in the Chebfun package [L. N. Trefethen and others, The Chebfun Development Team, 2011] and in the process generalized by the Chebfun authors to the roots of Jacobi, Laguerre and Hermite Polynomials. In this paper, we explore the generality of the link between barycentric weights and Gaussian quadrature and show that such relationships are related to the existence of lowering operators for orthogonal Polynomials. We supply an exhaustive list of cases, in which all known formulae are recovered and also some new formulae are derived, including the barycentric weights for Gauss-Radau and Gauss-Lobatto points. Based on a fast ${\mathcal O}(n)$ algorithm for the computation of Gaussian quadrature, due to Hale and Townsend, this leads to an ${\mathcal O}(n)$ computational scheme for barycentric weights.Comment: 23 pages, 4 figures, revised version with minor change

Puppo G. - One of the best experts on this subject based on the ideXlab platform.

  • CWENO: Uniformly accurate reconstructions for balance laws
    'American Mathematical Society (AMS)', 2018
    Co-Authors: Puppo G., Semplice M., Visconti G.
    Abstract:

    In this paper we introduce a general framework for defining and studying essentially nonoscillatory reconstruction procedures of arbitrarily high order of accuracy, interpolating data in the central stencil around a given computational cell (CWENO). This technique relies on the same selection mechanism of smooth stencils adopted in WENO, but here the pool of candidates for the selection includes Polynomials of different degrees. This seemingly minor difference allows us to compute the analytic expression of a Polynomial Interpolant, approximating the unknown function uniformly within a cell, instead of only at one point at a time. For this reason this technique is particularly suited for balance laws for finite volume schemes, when averages of source terms require high order quadrature rules based on several points; in the computation of local averages, during refinement in h-adaptive schemes; or in the transfer of the solution between grids in moving mesh techniques, and in general when a globally defined reconstruction is needed. Previously, these needs have been satisfied mostly by ENO reconstruction techniques, which, however, require a much wider stencil than the CWENO reconstruction studied here, for the same accuracy

  • CWENO: uniformly accurate reconstructions for balance laws
    'American Mathematical Society (AMS)', 2016
    Co-Authors: Puppo G., Semplice M., Visconti G.
    Abstract:

    In this paper we introduce a general framework for defining and studying essentially non-oscillatory reconstruction procedures of arbitrarily high order accuracy, interpolating data in a central stencil around a given computational cell ($\CWENO$). This technique relies on the same selection mechanism of smooth stencils adopted in $\WENO$, but here the pool of candidates for the selection includes Polynomials of different degrees. This seemingly minor difference allows to compute an analytic expression of a Polynomial Interpolant, approximating the unknown function uniformly within a cell, instead of only at one point at a time. For this reason this technique is particularly suited for balance laws for finite volume schemes, when averages of source terms require high order quadrature rules based on several points; in the computation of local averages, during refinement in h-adaptive schemes; or in the transfer of the solution between grids in moving mesh techniques, and in general when a globally defined reconstruction is needed. Previously, these needs have been satisfied mostly by ENO reconstruction techniques, which, however, require a much wider stencil then the $\CWENO$ reconstruction studied here, for the same accuracy

Wang Haiyong - One of the best experts on this subject based on the ideXlab platform.

  • Explicit barycentric weights for Polynomial interpolation in the roots or extrema of classical orthogonal Polynomials
    'American Mathematical Society (AMS)', 2014
    Co-Authors: Wang Haiyong, Huybrechs Daan, Vandewalle Stefan
    Abstract:

    © 2014 American Mathematical Society. Barycentric interpolation is arguably the method of choice for numerical Polynomial interpolation. The Polynomial Interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the interpolation points. Few explicit formulae for these barycentric weights are known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861-877], the authors have shown that the barycentric weights of the roots of Legendre Polynomials can be expressed explicitly in terms of the weights of the corresponding Gaussian quadrature rule. This idea was subsequently implemented in the Chebfun package [L. N. Trefethen and others, The Chebfun Development Team, 2011] and in the process generalized by the Chebfun authors to the roots of Jacobi, Laguerre and Hermite Polynomials. In this paper, we explore the generality of the link between barycentric weights and Gaussian quadrature and show that such relationships are related to the existence of lowering operators for orthogonal Polynomials. We supply an exhaustive list of cases, in which all known formulae are recovered and also some new formulae are derived, including the barycentric weights for Gauss-Radau and Gauss-Lobatto points. Based on a fast O(n) algorithm for the computation of Gaussian quadrature, due to Hale and Townsend, this leads to an O(n) computational scheme for barycentric weights.23 pages, 4 figures, revised version with minor changesstatus: publishe

  • Explicit barycentric weights for Polynomial interpolation in the roots or extrema of classical orthogonal Polynomials
    2012
    Co-Authors: Wang Haiyong, Huybrechs Daan, Vandewalle Stefan
    Abstract:

    Barycentric interpolation is arguably the method of choice for numerical Polynomial interpolation. The Polynomial Interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the interpolation points. Few explicit formulae for these barycentric weights are known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861--877], the authors have shown that the barycentric weights of the roots of Legendre Polynomials can be expressed explicitly in terms of the weights of the corresponding Gaussian quadrature rule. This idea was subsequently implemented in the Chebfun package [L. N. Trefethen and others, The Chebfun Development Team, 2011] and in the process generalized by the Chebfun authors to the roots of Jacobi, Laguerre and Hermite Polynomials. In this paper, we explore the generality of the link between barycentric weights and Gaussian quadrature and show that such relationships are related to the existence of lowering operators for orthogonal Polynomials. We supply an exhaustive list of cases, in which all known formulae are recovered and also some new formulae are derived, including the barycentric weights for Gauss-Radau and Gauss-Lobatto points. Based on a fast ${\mathcal O}(n)$ algorithm for the computation of Gaussian quadrature, due to Hale and Townsend, this leads to an ${\mathcal O}(n)$ computational scheme for barycentric weights.Comment: 23 pages, 4 figures, revised version with minor change

Moghaddam, Hassan Goldani - One of the best experts on this subject based on the ideXlab platform.

  • Applications of Generic Interpolants In the Investigation and Visualization of Approximate Solutions of PDEs on Coarse Unstructured Meshes
    2010
    Co-Authors: Moghaddam, Hassan Goldani
    Abstract:

    In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic Polynomial Interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise Polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based Interpolant and the contouring algorithm to three dimensions is also investigated. The use of the DEI-based Interpolant we introduce for visualization can also be used to develop effective Adaptive Mesh Refinement (AMR) techniques and global error estimates. In particular, we introduce and investigate four AMR techniques along with a hybrid mesh refinement technique. Our interest is in investigating how well such a `generic' mesh selection strategy, based on properties of the problem alone, can perform compared with a special-purpose strategy that is designed for a specific PDE method. We also introduce an \`{a} posteriori global error estimator by introducing the solution of a companion PDE defined in terms of the associated PCI.Ph